Discretization

Discretization

Choosing the imaginary-time discretization Dtau and managing Trotter errors. This applies to every ALF simulation.

Parameters

Set in the &VAR_Model_Generic namelist in the parameters file.

Parameter	Default	Description
Dtau	0.1	Imaginary-time step size
Beta	5.0	Inverse temperature ($\beta = 1/T$)

The number of imaginary-time slices is computed automatically:

$$L_\text{Trot} = \text{nint}(\beta / \Delta\tau)$$

For projective (zero-temperature) simulations with Projector = .T. and projection parameter Theta, the total number of slices is $L_\text{Trot} + 2 \times \text{nint}(\Theta / \Delta\tau)$.

Guidelines

Choosing Dtau

The Trotter decomposition introduces a systematic error of order $\mathcal{O}(\Delta\tau^2)$ in the partition function and observables. This error is not a statistical error — it is a bias that does not decrease with more sweeps or bins.

• Smaller Dtau = smaller Trotter error, but more time slices → more expensive.
• Larger Dtau = cheaper, but larger systematic bias.
• Dtau = 0.1 is a common starting point for moderate interaction strengths ($U/t \sim 1$–$4$).
• For strong coupling ($U/t > 6$) or high precision, reduce to Dtau = 0.05 or smaller.

Dtau Extrapolation

For publication-quality results, run at multiple Dtau values and extrapolate to $\Delta\tau \to 0$:

1. Run at e.g. Dtau = 0.2, 0.1, 0.05
2. Plot the observable vs. $\Delta\tau^2$
3. Fit a line and extrapolate to $\Delta\tau^2 = 0$

Since the error is $\mathcal{O}(\Delta\tau^2)$, a linear fit in $\Delta\tau^2$ is appropriate. If the results at your chosen Dtau already agree within error bars across different Dtau values, the Trotter error is negligible and extrapolation is not needed.

Interaction with Other Parameters

• Nwrap: The stabilization interval is Nwrap × Dtau. If you halve Dtau, you can double Nwrap and maintain the same stability. See Stabilization Parameters.
• Beta: Reducing Dtau at fixed Beta doubles Ltrot, roughly doubling the cost per sweep. The cost scales as $\mathcal{O}(L_\text{Trot} \times N_\text{dim}^3)$ for dense systems.
• Checkerboard decomposition: When Checkerboard = .T., the per-time-slice cost is reduced, making smaller Dtau more affordable.

Model-Specific Notes

• Hubbard model: Dtau = 0.1 is adequate for $U/t \leq 4$. For $U/t = 8$ or larger, consider Dtau = 0.05.
• Kondo model: The Kondo coupling can require finer discretization. Test convergence with Dtau explicitly.
• Models with continuous fields (HMC): Dtau affects both the Trotter error and the HMC force computation. Smaller Dtau generally leads to smoother forces and better HMC acceptance.

Known Pitfalls

• Dtau too large → biased results. Unlike statistical errors, Trotter errors do not average out. Results may look precise (small error bars) but be systematically wrong.
• Ltrot not an integer. ALF rounds $\beta / \Delta\tau$ to the nearest integer. Choose Dtau and Beta such that the ratio is close to an integer to avoid surprises (e.g., the actual Beta differs slightly from what you intended).
• Cost scaling. Halving Dtau doubles Ltrot and roughly doubles the wall time. Consider this before going to very small Dtau on large lattices.